Problem: $f(x, y) = 2x \ln(y) + 3$ $\dfrac{\partial^2 f}{\partial y \partial x} = $
Solution: Taking a mixed partial derivative is when we take two or more regular partial derivatives in a row, but each is with respect to a different variable. $\dfrac{\partial^2 f}{\partial y \partial x} = \dfrac{\partial}{\partial y} \left[ \dfrac{\partial f}{\partial x} \right]$ Let's differentiate! $\begin{aligned} \dfrac{\partial^2 f}{\partial y \partial x} &= \dfrac{\partial}{\partial y} \left[ \dfrac{\partial}{\partial x} \left[ 2x \ln(y) + 3 \right] \right] \\ \\ &= \dfrac{\partial}{\partial y} \left[ 2 \ln(y) \right] \\ \\ &= \dfrac{2}{y} \end{aligned}$ Therefore, $\dfrac{\partial^2 f}{\partial y \partial x} = \dfrac{2}{y}$.